Math Problem Statement
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. f(x) = {x + 1 if x < -2, -2x - 3 if x >= -2}
Solution
The problem involves sketching the graph of the piecewise function , which is defined as follows:
x + 1 & \text{if } x < -2, \\ -2x - 3 & \text{if } x \geq -2. \end{cases}$$ ### Step 1: Analyze Each Piece of the Function 1. **For $$x < -2$$**: The function is $$f(x) = x + 1$$, which is a linear equation with slope 1 and intercept 1. 2. **For $$x \geq -2$$**: The function is $$f(x) = -2x - 3$$, which is also linear but has a slope of -2 and a y-intercept of -3. ### Step 2: Graphing Each Piece - **First piece $$f(x) = x + 1$$ for $$x < -2$$**: - This is a straight line. We can calculate a couple of points to plot: - When $$x = -3$$, $$f(-3) = -3 + 1 = -2$$. - When $$x = -4$$, $$f(-4) = -4 + 1 = -3$$. - Since this part is valid only for $$x < -2$$, it will not include $$x = -2$$, and we'll use an open circle at the boundary $$x = -2$$. - **Second piece $$f(x) = -2x - 3$$ for $$x \geq -2$$**: - This is also a straight line. We can calculate points for this part: - When $$x = -2$$, $$f(-2) = -2(-2) - 3 = 4 - 3 = 1$$. - When $$x = 0$$, $$f(0) = -2(0) - 3 = -3$$. - This line includes $$x = -2$$, so the point $$(-2, 1)$$ will be a closed circle. ### Step 3: Domain in Interval Notation Since the function is defined for all real values of $$x$$, the domain is: $$\text{Domain} = (-\infty, \infty)$$ Would you like me to graph this piecewise function for you? Or provide more details? ### Related Questions: 1. How do you determine the boundary points for piecewise functions? 2. What happens when a piecewise function has a discontinuity at a certain value of $$x$$? 3. How can you verify if a piecewise function is continuous at the boundary point? 4. How does the slope of a linear equation affect the steepness of the line? 5. Can a piecewise function have non-linear pieces (such as quadratic or exponential)? ### Tip: When graphing piecewise functions, always pay attention to open and closed circles at the boundaries to avoid mistakes in representing which parts of the graph are included or excluded.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Formulas
f(x) = x + 1 for x < -2
f(x) = -2x - 3 for x >= -2
Theorems
Linear Function Properties
Piecewise Function Continuity
Suitable Grade Level
Grades 9-11